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Claim. The answer to [[problems/discrete_geometry/E1090/_index|Problem 1090]] is yes for $k=3$: some finite plane set has, in every two-coloring, a line through at least three of its points all of whose points in the set share one color. Erdős reports the result in his 1975 problem paper (card, Section 4, p. 106): after posing the question for every $k$, he writes that "GRAHAM and SELFRIDGE gave an affirmative answer for $k=3$, but the cases $k>3$ seem to be open." The paper gives neither a construction nor an argument, and no publication of the $k=3$ argument by Graham or Selfridge is recorded.

Covers. The case $k=3$ only. Every $k\geq3$ is settled on Hunter's claim page, whose construction gives the case $k=3$ as well.

Depends on. No page of this wiki.

Claimant. The result is Graham and Selfridge's; its only record is Erdős's report in Ann. Mat. Pura Appl. (4) 103 (1975), 99--108, an issue dated December 1975 with no day, so the page is dated to the publication month. The site's commentary (page last edited 2025-10-19) and the formal-conjectures statement file repeat the report in one sentence each and add nothing to it.

Acceptance. None recorded. Erdős's report is the poser's word that the case was answered, with no argument to examine, and this page does not count it as acceptance evidence. Thomas Bloom, the site's curator, labels the problem proved for Hunter's construction, which settles every $k$; that label credits nothing to this case. There is no refereed proof and no formalization of the Graham and Selfridge argument, so the claim stays claimed.